Chapter 1: Useful definitions 1.1. Cross-sections (review) The Nuclear and Radiochemistry class listed as a prerequisite is a good place to start. The understanding of a cross-section being fundamentai in this course I will review it quickly. 1.1.1. So what is a cross-section? The short story is : The cross-section represents the probability of an interaction. Now that is a quite good story but it lacks substance don t you think? Well I agree which is why I happily refer you to the previously cited class (chapter 3). As you thus already know a cross-section can be microscopic meaning that it is characteristic of an individual target or macroscopic meaning that it is characteristic of a material containing a large number of targets. 1.1.2. Microscopic cross-section It is very important to start by defining some of the vocabulary you ll have to know when talking of neutron-induced reactions. First scattering. This refers to any reaction that re-emits a neutron. This can be compared with a bouncing effect of the neutron on the atom. Second absorption. No surprise here ladies and gentlemen. This simply indicates that the neutron was absorbed by the atom it collided with. Now the absorption has two different consequences. It either induces the fission of the atom that absorbed the neutron i.e. the atom breaks into two smaller atoms and gives away neutrons or it induces a capture i.e. the atom accepts the additional neutron and is transformed into another atom an isotope of the first atom. In a mathematical form we can write : where and are respectively the fission and the capture cross-section.
Now all that means that we have a total microscopic cross-section that can be expressed as the sum of the absorption cross-section and the scattering cross-section. Indeed the absorption cross-section represents the probability that the neutron is absorbed (and then induces fission or capture) and the scattering cross-section the probability that the neutron is scattered. As those are the only options for the interactions it represents the total cross-section. Therefore we can write : where and are respectively the scattering and the absorption cross-section. The microscopic cross-section can be considered as an area of interaction. The larger the area the higher the probability of interaction. Note that the largeness of this area does not depend on the size of the atom itself no in fact it takes into account the energy of the incoming particle neutron in our case among other factors. As a sort of area the unit of a microscopic crosssection is. The area we re talking about being very small (of the order of ) physicists have created a specific unit the barn (b) to express those quantities.. 1.1.3. Macroscopic cross-section Now that we have seen the case of an individual target let s look at a higher level with a large number of targets. Imagine a neutron travelling in matter which the neutron sees as if it were a vacuum with nuclei here and there since the neutron is unaware of electrons. It therefore travels in a straight line at a constant speed until it hits a nucleus in its path very much like a flipper game. For a short path this collision probability is infinitesimal and proportional to the distance dx. This translates into : the path is twice as long? Then the probability of interaction is twice as high. This probability can be written as where is the appropriate proportionality coefficient called macroscopic cross-section. Neutron physicists generally use the centimetre as the unit of length for their calculations (order of magnitude for typical materials) and so macroscopic cross-sections are expressed in. Let us calculate the probability relationship that governs the distance point of the neutron and the point where it will have its first collision. between the starting So the first collision takes place between and + : if the neutron has had no collision between 0 and x This probability is denoted and if the neutron has a collision and + by definition this probability is The probability of the collision happening between and + is therefore : where is the probability density function. Now we need to calculate.
In order to do that we must realize that + ) -- the probability of no collision over the distance + -- is the product of : the probability of no collision between 0 and i.e. ; Indeed the neutron has to not collide before that point the probability of no collision between and + i.e. by definition. Indeed the neutron must not collide before + but is assumed to collide at + This gives : + We can notice that by definition. Indeed it is given that the neutron has not collided at its exact starting point because well we say so by defining 0 as the starting point. This allows us to simplify the previous equation and obtain : Question n 1 : Please demonstrate why. Rewriting the previous equation with the explicit expression of we find that : We have thus calculated the probability of the collision happening between and +. This allows us to calculate the mean free path of the neutrons i.e. the average value of the distance at which the first collision occurs. This is the inverse of the macroscopic crosssection : Question n 2 : Please demonstrate why. (Hint : the average value of the distance at which the first collision occurs ) Finally one can express the macroscopic cross-section as a function of the density of obstacles (nuclei) on the way and the probability of interacting with a specific obstacle (microscopic crosssection). Indeed the probability element of interaction with matter for a path is quite obviously proportional to the number of obstacles the neutron is likely to encounter and therefore to the number of nuclei per unit volume (# ). Once again one can try to visualize that by comparing with a flipper game. The proportionality coefficient is the probability of interaction with a specific target the microscopic cross-section ( ).
We can thus write : 1.2. Neutron density neutron flux reaction rate In a nuclear reactor the population of neutron is very small compared to the population of atoms (there is something like a factor!). But that does not mean that it is a small population in and by itself no. Indeed there are of the order of neutrons per cubic meter in a nuclear reactor. This is why we use the concept of density and statistic to describe their behavior. 1.2.1. So let s define the neutron density Imagine a small cube in the region of interest whose edges length is centered at a point. Its volume is therefore that is. Then consider you take a picture of that scene and look at it later (indeed everything is moving quite fast in there!). Then the average number of neutron found in this little cube is : where is the density (number of neutrons observed per unit volume). Keep in mind that this is an average. 1.2.2. Reaction rate Let s say that does not depend on (homogeneous repartition) for the sake of simplicity and let s call the speed of the neutrons. During a time interval a neutron travels a path. A very simple and well-known formula gives us : If we recall the previous section (macroscopic cross-section) we can remember that the probability of interaction of one neutron with matter over a path is. This then translates to : Note that the only thing done there was to inject the new expression of into the equation. If you want to know the number of neutron-matter interactions taking place in the little cube we were considering -- in more mathematical terms in the volume element -- we just have to multiply the probability that one neutron interacts with matter by the average number of neutrons in that volume. Thus the number of neutron-matter interactions taking place in the volume element during a time is : From that expression we can obtain what is called the reaction rate which is the number of interactions per unit volume and unit time :
1.2.3. Neutron flux If we look at the units of the product we can see that it is : thus apparently a number of neutron going through a surface area during a given time. This has the dimension of a flux doesn t it? Yes but it isn t a flux as we are used to see them. It has no connection to a surface it is a volumetric quantity. We ll go ahead and call that a flux anyway the angular flux. Why do this if it isn t an actual flux? Well that s a historical definition and things are really hard to change. In the previous equation I wrote. This corresponded to the total macroscopic cross-section. One can notice that we can separate this total macroscopic cross-section into several interaction cross-sections ( for absorption for scattering) which yields to a specific reaction rate per interaction. Just wanted to clarify that. Thus we finally obtain : where represents the matter (as in the cross-section depends on the matter density and other variables ) and represents the neutron population. 1.3. The phase space We have to use seven independent variables to describe the neutron population behavior in space and time : Space : Velocity : Time : One has to notice that this Cartesian coordinate system can be translated into a spherical one (you can see the translation graph on figure 1)
fig. 1 The materials inside the reactor are theoretically isotropic which means that no matter which angle they re seen from they have the same properties. Cross-sections are thus not dependent on the direction of the incident neutron but only on its speed and by speed I mean energy. Indeed instead of velocity we use kinetic energy ( ) and direction ( ) : with the unit velocity vector identified by the two angles of the spherical coordinates : We can keep in mind several useful formulas concerning : So that s all good and stuff but what next? Well this takes us to the notion of angular density the that we have talked about earlier. When I introduced this density I did not consider speed nor time. This angular density depends on. So is the number of neutrons in the volume around and with their velocity in the volume (velocity space) around at time. If as we said earlier we use and instead of we obtain instead of.
Then the number of neutrons in the volume around with energy between and and with their direction between and at time is : Consequently the relationship between and is given by the change of variables : Let us quickly come back to the fluxes for a moment. After all it will be the main focus of our work later on. We ve seen that we can write the angular flux (not a real flux) as the product of the neutron density and their velocity i.e. : From there it is also possible to define what we ll call the scalar flux. The scalar flux is the angular flux integrated over all angles : In a similar way integrating over all angles the angular density gives us the energy dependent density : 1.4. Concept of current We ve seen that neutron physicists used the term flux to describe something that despite having the unit of a flux was not a flux. And they ve been consistent in this notation changes. That s why the term current is used in neutronics to denote what is known as flux in other branches of physics : the number of neutrons passing through a surface area normalized per unit surface and unit time. 1.4.1. Angular current density We can first define the angular current density which is the angular flux multiplied by the unit velocity vector.
1.4.2. Neutron current Now consider a surface element with surface area and located perpendicular to the unit vector (normal). Let s examine the neutrons with a given direction between and. The situation can be seen on figure 2 below. fig. 2 We can define the neutron current i.e. the number of neutrons with energy which cross the surface per unit time : and direction 1.4.3. Current density As with the angular and scalar flux one can integrate the angular current density over all angle in order to obtain the current density.
1.4.4. Partial currents fig. 3 The current considers both neutrons going in a direction (following ) and in the opposite direction ( ). Therefore we can further define partial currents and. It is useful to note that indicates that the direction of the neutron projected on the normal is positive. This means that if we look at figure 3 (or figure 2) the neutron with such direction is going from left to right hence. Of course represents the opposite hence. If the flux is isotropic i.e. if it doesn t have a privileged direction (independent of ) then : Question n 3 : Please demonstrate why in the case of an isotropic flux.
Well this ends the first lecture. If you have any question please let me know directly or post a thread in the dedicated subreddit. Do not forget and I can t stress this enough : if you have a question then someone else in the class is wondering the same thing or should be. Therefore asking it will help you and others. The homework #1 (which will be the three questions that I ve written in this lecture in order to give you context as well as a bonus problem) can be difficult if you are not familiar with the mathematics involved. If you have any difficulty (it s definitely okay not to know how to integrate a hidden multiple integral) once again use the discussion platform (refer to previous link) or email me directly. There is another thing that I should say. If you do not understand something do not feel like it s your fault and do not give up. It merely means that my explanations were not good enough. I will gladly upgrade the class by taking into account your suggestions and remarks.